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I am trying to learn derivates again. And was going over this calculation

https://www.mathway.com/popular-problems/Calculus/531323

It confused me :

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I thought derivative of log(x) with respect to x is 1/x.

Source: https://simple.wikipedia.org/wiki/Derivative_(mathematics)

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    That's for ln(x) – 2017-01-08
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    The derivative of the Naperian log is $u \mapsto {1 \over u}$, the derivative of the base 10 log follows from the fact that $\log_{10} u = {\ln u \over \ln 10}$. – 2017-01-08

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Mathematicians writing $\log u$ almost always mean the base-$e$ logarithm of $u$, and then one has $$\frac d {du} \log u = \frac 1 u.$$ Calculators almost always take $\log$ to mean base-$10$ logarithm, and one has $$\frac d {du} \log_{10} u = \frac 1 {u \log_e 10}.$$ The convention that $\log$ means the base-$10$ logarithm is followed in some contexts in some disciplines in science and engineering, but is not generally followed in mathematics.

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If your $\log x$ is $\ln x,$ then what you said is correct.

However, some regard $\log x$ as $\log_{10} x.$ In this case you will have $$(\log u)'=\left(\frac{\ln u}{\ln 10}\right)'=\frac{1}{u\ln 10}.$$

I think since in your reference, they had used $\ln$ to denote natural log, then $\log$ may be used to mean above.

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You're confusing the derivative of the natural logarithm, and the logarithm in base $10$. For the natural logarithm we have: $$\frac{d}{du}\log_e(u)=\frac{d}{du}\ln(u)=\frac{1}{u}\qquad u> 0$$ Now for the base $10$ logarithm (often simply denoted $\log(u)$), we use the base conversion formula to express the logarithm in terms of the natural logarithm (base $e$), and then we differentiate. We have: $$\frac{d}{du}\log_{10}(u)=\frac{d}{du}\log(u)=\frac{d}{du}\frac{\ln(u)}{\ln(10)}=\frac{1}{\ln(10)u}\qquad u> 0$$

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    You should also have mentioned that the base-$e$ logarithm is also often denoted simply $\log$ rather than $\log_e. \qquad$ – 2017-01-08
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    @MichaelHardy I see the base $e$ logarithm denoted $\ln$, and the base $10$ logarithm denoted $\log$, but haven't seen the base $e$ logarithm denoted $\log$ (personally haven't seen this). – 2017-01-08
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    If you take any theoretical math course, as opposed to math courses taken by students who are there in order to get grades to impress medical schools to which they apply for admission, you will see this. Also, in most programming languages, $\log$ means natural logarithm. – 2017-01-08